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Congruent Circles Theorem

Rule

Congruent Circles Theorem

Two circles are congruent circles if and only if they have the same radius.

Proof

Consider two congruent circles and and a point on each one.

Because the distance from the center of the circle to a point on the circle is the same for each circle. Therefore, which implies that both circles have the same radius.

Conversely, consider two circles with the same radius.

Next, translate so that point maps to point The image of is which is a circle centered at Since the circles have the same radius, this translation maps onto
Translate

Reset

Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.